Problem 159 - College Algebra

Question

How is the quadratic formula derived?

Step-by-Step Solution

Answer

The quadratic formula is derived from the standard formula of a quadratic equation by making the coefficient of \(x^2\) equal to 1, expressing the equation in a square form, simplifying and isolating 'x' to find the roots. The formula is \(x = [ -b ± sqrt( b^2 - 4ac ) ] / 2a\)

Step 1: Understanding the Standard Quadratic Equation

A quadratic equation is usually written in the form \(ax^2 + bx + c = 0\), where a, b and c are constants, 'a' not equal to 0, 'x' is the variable.

Step 2: Divide by 'a'

Divide through by 'a' to make the coefficient of \(x^2\) equal to 1. The equation becomes \(x^2 + (b/a)x + c/a = 0\).

Step 3: Express in square form

Express the equation in a square form. This can be done by adding and subtracting \((b/2a)^2\) to the left-hand side of the equation, and keeping the right-hand side equal to 0. So, \((x + b/2a)^2 - (b/2a)^2 + c/a = 0\)

Step 4: Adjust right hand side

Move \((b^2/4a^2)\) to the right-hand side of the equation, leaving \(x + b/2a)^2\) in the left-hand side of the equation. So, \((x + b/2a)^2 = (b^2/4a^2) - c/a\)

Step 5: Simplification

Simplify the right-hand side of the equation, you get, \((x + b/2a)^2 = (b^2 - 4ac)/(4a^2)\)

Step 6: Take the square root on both sides

Take the square root on both sides of the equation, you'll get \(x + b/2a = ± \sqrt{ (b^2 - 4ac) / 4a^2 }\)

Step 7: Isolate x

The last and final step is to isolate 'x' in order to find the roots of the quadratic equation. So, \(x = [ -b ± sqrt( b^2 - 4ac ) ] / 2a\) This is called the Quadratic Formula.